The Hashiwokakero Puzzle is a logical puzzle where one has to build **bridges** between islands.
The puzzle is also known under the name **Ai-Ki-Ai**.
The puzzles can also be played online.

The requirements for this puzzle are as follows:

- the goal is to build bridges between islands so as to generate a
**connected**graph - every island has a number on it, indicating exactly how many bridges should be linked with the island
- there is an upper bound (MAXBRIDGES) on the number of bridges that can be built-between two islands
- bridges cannot cross each other

A B model for this puzzle can be found below. The constants and sets of the model are as follows:

- N are the nodes (islands); we have added a constant ignore where one can stipulate which islands should be ignored in this puzzle
- nl (number of links) stipulates for each island how many bridges it should be linked with
- xc, yc are the x- and y-coordinates for every island

A simple puzzle with four islands would be defined as follows, assuming the basic set N is defined as `N = {a,b,c,d,e,f,g,h,i,j,k,l,m,n}`:

xc(a)=0 & xc(b)=1 & xc(c)=0 & xc(d) = 1 & yc(a)=0 & yc(b)=0 & yc(c)=1 & yc(d) = 1 & nl = {a|->2, b|->2, c|->2, d|->2} & ignore = {e,f,g,h,i,j,k,l,m,n}

The model then contains the following derived constants:

- plx,ply: the possible links between islands on the x- and y-axis respectively
- pl: the possible links both on the x- and y-axis combined
- cs: the conflict set of links which overlap, i.e., one cannot build bridges on both links (a,b) when the pair (a,b) is in cs
- connected: the set of links on which at least one bridge was built

The model also sets up the goal constant `sol` which maps every link in `pl` to a number indicating how many bridges are built on it.
The model also stipulates that the graph set up by connected generates a fully connected graph.

Here is the full model:

MACHINE Bridges DEFINITIONS MAXBRIDGES==2; LINKS == 1..(MAXBRIDGES*4); COORD == 0..10; p1 == prj1(nodes,nodes); p2 == prj2(nodes,nodes); p1i == prj1(nodes,INTEGER) SETS // N = {a,b,c,d} N = {a,b,c,d,e,f,g,h,i,j,k,l,m,n} CONSTANTS nodes, ignore, nl, xc,yc, plx,ply,pl, cs, sol, connected PROPERTIES nodes = N \ ignore & // target number of links per node: nl : nodes --> LINKS & /* number of links */ // coordinates of nodes xc: nodes --> COORD & yc: nodes --> COORD & // possible links: pl : nodes <-> nodes & plx : nodes <-> nodes & ply : nodes <-> nodes & plx = {n1,n2 | xc(n1)=xc(n2) & n1 /= n2 & yc(n2)>yc(n1) & !n3.(xc(n3)=xc(n1) => yc(n3) /: yc(n1)+1..yc(n2)-1) } & ply = {n1,n2 | yc(n1)=yc(n2) & n1 /= n2 & xc(n2)>xc(n1) & !n3.(yc(n3)=yc(n1) => xc(n3) /: xc(n1)+1..xc(n2)-1)} & pl = plx \/ ply & // compute conflict set (assumes xc,yc coordinates ordered in plx,ply) cs = {pl1,pl2 | pl1:plx & pl2:ply & xc(p1(pl1)): xc(p1(pl2))+1..xc(p2(pl2))-1 & yc(p1(pl2)): yc(p1(pl1))+1..yc(p2(pl1))-1} & sol : pl --> 0..MAXBRIDGES & !nn.(nn:nodes => SIGMA(l).(l:pl & (p1(l)=nn or p2(l)=nn)|sol(l))=nl(nn)) & !(pl1,pl2).( (pl1,pl2):cs => sol(pl1)=0 or sol(pl2)=0) // no conflicts & // check graph connected connected = {pl|sol(pl)>0} & closure1(connected \/ connected~)[{a}] = {nn|nn:nodes & nl(nn)>0} // encoding of puzzle & // A puzzle from bridges.png xc(a)=1 & yc(a)=1 & nl(a)=4 & xc(b)=1 & yc(b)=4 & nl(b)=6 & xc(c)=1 & yc(c)=6 & nl(c)=3 & xc(d)=2 & yc(d)=2 & nl(d)=1 & xc(e)=2 & yc(e)=5 & nl(e)=2 & xc(f)=3 & yc(f)=2 & nl(f)=4 & xc(g)=3 & yc(g)=4 & nl(g)=6 & xc(h)=3 & yc(h)=5 & nl(h)=4 & xc(i)=4 & yc(i)=3 & nl(i)=3 & xc(j)=4 & yc(j)=6 & nl(j)=3 & xc(k)=5 & yc(k)=2 & nl(k)=1 & xc(l)=6 & yc(l)=1 & nl(l)=4 & xc(m)=6 & yc(m)=3 & nl(m)=5 & xc(n)=6 & yc(n)=5 & nl(n)=2 & ignore = {} END

A solution for the above Puzzle (in Bridges.png) is found by ProB in 0.08 seconds (on a MacBook Air 2.2GHz i7).

To show the solution graphically, we can add the following to the `DEFINITIONS` clause in the model:

CUSTOM_GRAPH_NODES == {n,w,w2|(n|->w):nl & w=w2}; // %n1.(n1:nodes|nl(n1)); CUSTOM_GRAPH_EDGES == {n1,w,n2|n1:nl & n2:nl & (p1i(n1),p1i(n2),w):sol}

One can then load the model, perform the initialisation (double clicking on

INITIALISATION

in the operations pane) and the execute the command "Current State as Custom Graph" in the States sub-menu of the Visualize menu. This leads to the following picture: